The bifurcating closed orbits correspond to fixed points of the Poincare map

(1)

ON HOPF BIFURCATION OF RAYLEIGH'S EQUATION M. FARKAS, L. SPARING and Gy. SZAB6

Mechanical Engineering, Department of Mathematics.

Technical University. H-1521 Budapest, Received August 28. 1984

Summary

Rayleigh's differential equation exhibits a supercritical Hopfbifurcation. The bifurcating closed orbits correspond to fixed points of the Poincare map. In this paper explicit estimates of the domain of the Poincare map are given and these estimates are checked by a computer. The results may help the estimate of the possible amplitude of the stable oscillation of this important nonlinear system.

Introduction

Rayleigh's differential equation

d

2

U+dU((du))"2 1

1

)+11=0, pER

dt

²

dt dt

⁽¹⁾

is an important model used extensively in the theory of non-linear oscillations (see e. g. Minorsky [3J). A system equivalent to the second order scalar equation (I) is

(2) where dot denotes differentiation with respect to t ER. This system satisfies all the conditions of the classical Hopf bifurcation theorem [1

J

(or see Marsden- McCracken [2J). That is to say, the origin x=(x_{1 ,}x₂)=0 is an equilibrium of system (2) for all 1I E R. For 1I <

°

it is asymptotically stable, for p >

°

^{it is}

unstable. A 6>

°

and a smooth function 11: ( - 6, (»)f----7 R exist such that prO)

=

0,

1/(0)

= °

and for all x? E ( - (), ()) the solution of

corresponding to the initial condition (x?, 0) is periodic, and in a three dimensional neighbourhood of (x l' X 2' It) = (0,0,0) the corresponding trajec- tories arc the only closed paths of system (2). Moreover. applying the method of

(2)

264 .If. FARKAS (.[ af.

Negrini-Salvadori [4J a Lyapunov function

2 2 3 3 5 3

F(X 1,X²)=X 1 +x²+ 4X1X2+ 4 X1X2

can be determined which is, obviously, positive definite in a neighbourhood of the origin x

= °

and its derivative with respect to system (2) with It

= °

^is

negative definite: 3

F(x1 , X 2 ) = - 4 (x~

+

X~)2

+

e(lxI⁴).

This implies that x =

°

is a 3-attractor (a vague attractor) of system (2) with fJ.

= °

and, as a consequence, fJ.(x 1) > 0, ^Xl:;6 0, and the bifurcating closed paths are orbitally asymptotically stable.

In this paper we are giving an explicit extimate of the domain of the Poincare map attached to the problem. This estimate may enable us to estimate either the domain or the range of the function fJ., i. e. the values of the bifurcation parameter fJ. and the initial conditions x? for which still bifurcating closed paths exist. (It is, of course, known that (1) has a non-constant periodic solution for all fJ.:;6 0, since, (1) is equivalent to van der Pol's equation if fJ.:;6 0). Our method seems to be fairly general and possibly applicable in treating bifurcations of other two dimensional sy<;tems. However, the estimates gained by this exact method are rather c( nservative compared to results gained by computer experiments. In section 2 the domain of the Poincare map is estimated from below, and in section 3 the results gained by a computer are presented.

The Estimate of the Poincare Map

First of all, system (2) will be transformed by the polar transtormation

Xl =r cos

e,

_{X 2}=r sin

e

into

r=r

sin²e(fJ.-r2 sin²e)

e ⁼

^-1- ^r₂2 ^sin^2e^sin²^e

⁺ I

^sin^2e. ⁽³⁾

The solution of (3) assuming the initial values (r, e) at t

= °

will be denoted by (f(t, r, e, fJ.), 8(t, r, e, fJ.)). It will be assumed always that r"2 0. If the initial values are (0, e) then, clearly, f(t, 0, e, fJ.)=0 and the function 8(t, 0, e, fJ.) satisfies

.:.. fJ.. -

e(t, 0, e, fJ.)= -1

+ "2

^sIn2e(t,0, e, fJ.). (4) In particular, 8(t, 0, 0, 0) = - t and 8(21[,0,0,0) = - 21[. As a consequence, it is clear that for small enough fJ. and Initial condition ^(r,0) the integral curves of the

(3)

265

solutions of (3) will cut the

e

^{= -} ²ⁿplane (in t, r,

e

space) at some moment

t = T(p, r) > 0, see Fig. 1.

I n view of the polar transformation the initial .::ondition (x ^l'x ²⁾

=

^{(r, 0)}

attached to system (2) corresponds to the initial condition (r, 0) attached to (3).

21[ T(.u. r )

---

---

Fig. I. Integral curves of the solutions of (3) ::>:: f.L=0, (r, 8)=(0,0);

fJ: f.L smalL r small, 8 =

°

If (<P1(t, _{X 1 ,X 2},P), <P2(t, Xl' x2 , p)) denotes the solution of (2) satisfying the initial conditions

(<p ¹(0, Xl' X 2' Il), <P2 (0, Xl' X 2, p)) = (X 1 , x ²⁾then

<PI (T(p, r), r, 0, p)=f(T(p, r), r, 0, p) cos (-2n)=

=

^f(T(p,r), r, 0, p) 2:: 0,

<P2(T(p, r), r, 0, p)=f(T(p, r), r, 0, p) sin (-2n)=0.

This means that the path corresponding to the solution (<Pl(t, r, 0, p), <P2(t, r, 0, p)) of (2) intersects the positive ^Xlaxis at moment t= T(p, r) the first time after the intersection (r, 0) at t

=

0. See Fig. 2. The Poincare map, the domain of which we want to estimate, is the mapping defined by

(p, r)f--+ <P 1 (T(p, r), r, 0, p). (5) In order to get an estimate for the domain of this mapping we have to establish an estimate for the solutions of (3).

(4)

266 If. f-"AR};AS "f <1/.

LEMMA. The solution (f(t, r, 0, ^fl),8(t, r, 0, ^fl))of (3) corresponding to the initial condition (r, 0) r> 0 and to the parameter value J-i:2: 0 is defined in

Os t < 00, and satisfies the inequality

O<f(t, r, 0, fl)sr(l +tfl2/2r2)1/2. (6)

x,

X,

o

Fig, 2. Poincare map attached to (2) around the origin

PROOF. If r > 0 the corresponding path cannot cross the axis r

=

0 since the latter is itself a trajectory, thus, f(t, r, 0, ^fl)> 0 in its existence domain. Taking this into account we estimate the maximum of the function standing on the right hand side of the first equation of (3) for fixed r > 0 and fl ~ 0:

After a somewhat lengthy but elementary calculation we get

Here the right hand side is the actual value of the maximum if 0 ^S^fl^S^2r2and it is an upper estimate in case J1 > ^2r2.Thus, the following inequality holds

Solving this differential inequality we get (6) for t? 0 in the existence domain of the solution. The right hand side of (6) is bounded in every bounded interval.

Thus, f is bounded in every bounded interval, and in view of the second

(5)

ON HOPF BIFURCATION OF RA YLElGHS EQUATlOS 267

equation of(3), the same is true for

19

and, as a consequence, for

e.

This implies that the solution is defined in [0, CI)), and this completes the proof.

As it was shown at the beginning of this section, the integral curve of (3) corresponding to the initial condition (0, 0) and to the parameter value J1. =

°

cuts the 8

= -

2n plane at t

=

2n. Therefore, we are going to fix an interval larger than [0, 2n] to ensure a crossing of the same plane by the integral curve corresponding to the initial condition (r, 0), r:?:

°

and the parameter value J1.:?: 0.

THEOREM. Let us choose a constant a> 1 and let J1. and r satisfy the inequalities

(7) then the integral curve of the solution (1';(t, r, 0, J1.), e(t, r, 0, J1.)) cuts the plane 8 =

- 2n at some moment T(J1., r) E (0, 2an].

PROOF. We get from the second inequality of (7) that

for t E [0, 2an], r > 0. Hence applying (6) we get that p2(t, r, 0, J1.) ~ 2(a - 1 )/a

for t E [0, 2an], r > 0, J1.:?: 0. However, the last inequality is trivially true also for r 0. The first inequality of(7) and the one we have just obtained imply (the writing out of the arguments will be omitted)

.:.. 1 ? ? _ _

8(t, r. 0, J1.)= -1

+

2 (p-r- sin-8) sin 28 ~

1 1

1

^{- ? ?}

-I

^a-I ¹

~

- +?

^p-^r-^{sm- 8} ~ -1

+ - - = - - .

- a a

Integrating from

°

^to^2an^{we get} e(2an, r, 0, /-l) ~ - 2n. Thus, for some 0< T(J1., r) ~ 2an we have e(T(J1., r), r, 0, p) = - 2n and this was to be proved.

COROLLARY. For any fixed a> 1 the set Da defined by (7) is a subset of the domain of the Poincare map (5).

(6)

268 .If. FARKAS <', ul.

Numerical calculations show that the optimal set Da is obtained if a = 2.5 is chosen. Introducing the notation

F_a(Jl)=(2(a-l)/a- anp^2)1/2

Figure 3 shows the set D_{2 . 5}and the graph of the function Jl discussed in the Introduction. The latter has been gained by numerical methods.

J

OL-____________ ^~~______ ^~~__ ^~~____ ^~

032 0.39 j.J

0.1

Fil!. 3. Estimate of the domain of the Poincare map

Numerical Results

The fixed points of the Poincare map defined in (5) can be determined by computer. To each r> 0 small enough the value p (r) of the function p can be uniquely determined for which r= CfJdT(p(r), r), r, 0, p(r)) holds. The corresponding solution (CfJl(t, r, 0, p(r)), CfJ2(t, r, 0, p(r)) of(2) (where p=Jl(r)) is periodic with period T(p(r), r).

We have used a second generation computer of type ODRA-1204 and a elL digital plotter to get graphical results. A Runge-Kutta type method was applied to solve system (2). We could bring down the relative error of the Poincare map below 10 -5, keeping, at the same time, computer time within reasonable limits. We have determined the values of the function p and the corresponding periods T(p(r), r) first within the limits of the domain established

--- - - - - -_ .. _----_._-_. -_.-._ ... -

(7)

O,V 1/01'1' BII'( ReAT/OS OF RA ),I.U(;IIS ICQLITlOS 269

theoretically (Fig. 3) then the values of r were increased considerably and the closed paths remained to exist, though became more and more deformed. The results are shown in Table I. It can be seen from the fourth column that for small values of r we have Il(r):::::0.75r2.

Table 1

Initial value (r). corresponding bifurcation parameter value II(r), period T(IL(r), r) and II(r)/r²for Rayleigh's equation

/£(r) T(/I(r), r) p(r)/r²

.1 .om 503 6.283207 0.7503

.2 .029999 6.283538 0.7500

J .067451 6.2R4970 0.7495

.4 .119699 6.288809 0.7481

.5 .186359 6.296 ROR 0.7454

.6 .266640 6.311050 0.7407

.7 .359269 6.333690 0.7332

.8 .462407 6.366648 0.7225

.9 .573786 6.41 1260 0.7084

1.0 .690961 6.468034 0.6910

LI .81 I 566 6.536639 0.6707

1.2 (.J33595 6.616046 0.6483

L3 L05549X 6.704795 0.6246

1.4 L176237 6.801289 0.6001

1.5 1.295205 6.903970 0.5756

2.0 1.857 I11 7.469813 0.4643

2.5 2.370079 X.065658 0.3792

3.0 2.846108 8.660461 0.3162

3.5 3.293772 9.2451XO 0.2689

4.0 3.718595 9.817202 0.2324

4.5 4.124565 10.376058 0.2037

5.0 4.514543 10.922108 0.1806

6.0 5.254925 I L97X 473 0.1460

7.0 5.952409 12.992 099 0.1215

8.0 6.615321 13.968404 0.1034

9.0 7.249579 14.911949 0.0895

lO.O 7.X59517 15.826517 0.07R6

Figure 4 shows some of the bifurcating closed paths from r = 0.1 to r = 1.5.

Figure 5 shows the graphs of the functions n--;. tl(r) and rf--7 T(fl(r), r). One can see from Figure 6 how strongly attractive is the closed path corresponding to r = 1, fl{ 1 ) = 0.69096 yet.

(8)

270 ^\I.FA RKA,,)' et ai.

x,

Fill. 4. Bifurcating closed paths of Rayleigh's equation

t Y 10

15 l=Tlplrl.rl

10 r Fill. 5. Graphs of the functions n--+ p(r) and n ... T(p(r).r)

(9)

ON HOP!' Bln"RCATlON OF RA )'LE/GfI'S EQUATION 271

X,

Fig. 6. The attractive closed path of Rayleigh's equation for J1 = 0.69096

References

1. HOPF, E.: Abzweigung einer periodischen Losung von einer stationaren Losung eines Differentialsystems, Ber. Math.-Phys. Sachsische Akad. Wiss. Leipzig, 94 I. (1942).

2. MARSDEN,1. E.-McCRACKEN, M.: The Hopfbifurcation and its applications, Springer, New York, 1976.

3. MINORSKY, N.: Introduction to non-linear mechanics, Edwards, Ann Arbor, 1947.

4. NEGR!N!, P.-SALVADOR!, L.: Attractivity and Hopf bifurcation, Nonlin. Anal. T. M. A. 3 87.

(1979)

Prof. Dr. Miklos F ARKAS

I

L<iszlo SPARING, H-1521, Budapest Dr. Gyorgy Szabo